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Article

Underlying Geometric Flow in Hamiltonian Evolution

by
Gil Elgressy
1,* and
Lawrence Horwitz
1,2,3,*
1
Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
2
Department of Physics, Ariel University, Ariel 44837, Israel
3
School of Physics, Tel Aviv University, Ramat Aviv 69978, Israel
*
Authors to whom correspondence should be addressed.
Entropy 2025, 27(5), 510; https://doi.org/10.3390/e27050510
Submission received: 20 March 2025 / Revised: 25 April 2025 / Accepted: 2 May 2025 / Published: 9 May 2025
(This article belongs to the Special Issue Unstable Hamiltonian Systems and Scattering Theory)

Abstract

In this paper, an underlying perturbed Ricci flow construction is made within the metric operator space, originating from the Heisenberg dynamical equations, to formulate a method which appears to provide a new geometric approach for the geometric formulation of the quantum mechanical dynamics. A quantum mechanical notion of stability and local instability is introduced within the quantum mechanical theory, based on the quantum mechanical dynamical equations governing the evolution of the tensor metric operator. The stability analysis is conducted in the topology of little Ho¨lder spaces of metrics which the tensor metric operator acts on. Finally, a theorem is introduced in an attempt to characterize the stability properties of the quantum mechanical system such that it brings the quantum mechanical dynamics into the analysis.
Keywords: Hamiltonian evolution; tensor metric operator; underlying geometric flow; adiabatic flow; metric operator space; quantum mechanical Ricci flow; Ho¨lder spaces; convergence stability; local instability Hamiltonian evolution; tensor metric operator; underlying geometric flow; adiabatic flow; metric operator space; quantum mechanical Ricci flow; Ho¨lder spaces; convergence stability; local instability

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MDPI and ACS Style

Elgressy, G.; Horwitz, L. Underlying Geometric Flow in Hamiltonian Evolution. Entropy 2025, 27, 510. https://doi.org/10.3390/e27050510

AMA Style

Elgressy G, Horwitz L. Underlying Geometric Flow in Hamiltonian Evolution. Entropy. 2025; 27(5):510. https://doi.org/10.3390/e27050510

Chicago/Turabian Style

Elgressy, Gil, and Lawrence Horwitz. 2025. "Underlying Geometric Flow in Hamiltonian Evolution" Entropy 27, no. 5: 510. https://doi.org/10.3390/e27050510

APA Style

Elgressy, G., & Horwitz, L. (2025). Underlying Geometric Flow in Hamiltonian Evolution. Entropy, 27(5), 510. https://doi.org/10.3390/e27050510

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